Good problems in Algebraic Geometry

I am now using Fulton's book Algebraic Curves to learn algebraic geometry from and have just finished chapter 2. However I feel that the problems are not very inspiring (at the moment at least) and lack some depth. Where is a good source of problems in algebraic geometry that I can find at least at the level of Fulton?

I don't mind if people recommend specific problems from Hartshorne say as long as I can do them with the tools I have from Fulton. To be more specific, the next chapter of Fulton is on local properties of plane curves and computing intersection numbers, so if one can recommend problems for these, it would be good.

Thanks.

Thomas A. Garrity has led a remarkable project aiming at teaching elementary algebraic geometry by means of problems.
Definitions are given but there are no proofs. The readers, meticulously guided by a progression of exercises which constitute the heart of the course, should find the proofs by themselves: the Moore method in all its splendour!
The level is more elementary than Fulton's.

This project has very recently become the AMS book Algebraic Geometry: A Problem Solving Approach.
I haven't seen that book, but I believe that my first link above is to the manuscript used for the published book.

Evan Bullock's page for his Introduction to Algebraic Geometry might interest you.

Apart from 11 homeworks containing 7 or 8 exercises each, you'll find on it several interesting hand-outs and excerpts on Veronese, Segre, Grassmannians, symmetric and alternating tensors, dimension, ... by him or outstanding mathematicians like Harris, Atiyah-Mcdonald and others.
Bullock's course has as accompanying textbook Shafarevich's Basic Algebraic Geometry , which is at about the same level as Fulton: both are favourite introductions to algebraic geometry.

The exercises fit beautifully with Fulton. Here is an example:
In exercise 2.34 Fulton asks you to show that the sum $f=f_{d-1}+f_d$ of two homogeneous polynomials of degrees $d-1,d $ in $n$ variables (without common factors) is irreducible.
And Bullock in his first mid-term examination asks you to show that the zero locus $ V(f)\subset \mathbb A^{n}$ of $f$ is birational to $\mathbb A^{n-1}$.
The icing on that particular cake is that you will find a solution to this exercise (and to others) here.